Atomic spectra

Spectroscopy of Atoms¶

• Spectroscopy is the study of the interaction between matter and electromagnetic radiation.

• By analyzing the emitted or absorbed light, spectroscopy reveals information about the structure and composition of atoms and molecules.

• When heated or subjected to electrical discharge, atoms emit radiation at characteristic frequencies. The resulting spectrum is unique for each element, serving as a kind of atomic fingerprint.

Every element produces its own set of lines. Below is what a spectrograph records from a discharge lamp of each gas: no two patterns are alike, which is why a spectrum taken through a telescope tells you what a star is made of.

Source

import numpy as np
import matplotlib.pyplot as plt

def wl_to_rgb(wl):
    """Approximate sRGB colour of a single wavelength in nm."""
    if wl < 440:    r, g, b = -(wl - 440) / 60, 0.0, 1.0
    elif wl < 490:  r, g, b = 0.0, (wl - 440) / 50, 1.0
    elif wl < 510:  r, g, b = 0.0, 1.0, -(wl - 510) / 20
    elif wl < 580:  r, g, b = (wl - 510) / 70, 1.0, 0.0
    elif wl < 645:  r, g, b = 1.0, -(wl - 645) / 65, 0.0
    else:           r, g, b = 1.0, 0.0, 0.0
    return (min(r, 1.0), min(g, 1.0), min(b, 1.0))

# (wavelength in nm, relative brightness) for the strong visible lines
spectra = {
    "H":  [(656.3, 1.0), (486.1, 0.6), (434.0, 0.4), (410.2, 0.3)],
    "He": [(447.1, 0.5), (471.3, 0.3), (492.2, 0.3), (501.6, 0.6),
           (587.6, 1.0), (667.8, 0.5), (706.5, 0.4)],
    "Na": [(498.3, 0.2), (568.8, 0.3), (589.0, 1.0), (589.6, 1.0), (615.4, 0.3)],
    "Hg": [(404.7, 0.6), (435.8, 1.0), (546.1, 1.0), (577.0, 0.6), (579.1, 0.6)],
    "Ne": [(585.2, 0.8), (588.2, 0.6), (594.5, 0.7), (607.4, 0.5), (614.3, 0.7),
           (621.7, 0.5), (626.6, 0.6), (633.4, 0.8), (640.2, 1.0), (650.7, 0.6),
           (659.9, 0.5), (692.9, 0.4), (703.2, 0.5)],
}

fig, axes = plt.subplots(len(spectra), 1, figsize=(9, 4.0), sharex=True)
for ax, (name, lines) in zip(axes, spectra.items()):
    ax.set_facecolor("black")
    for wl, intensity in lines:
        ax.axvline(wl, color=wl_to_rgb(wl), lw=1.8, alpha=0.35 + 0.65 * intensity)
    ax.set_yticks([])
    ax.set_ylabel(name, rotation=0, ha="right", va="center", fontsize=12, labelpad=14)
    for spine in ax.spines.values():
        spine.set_visible(False)
axes[-1].set_xlim(380, 750)
axes[-1].set_xlabel("wavelength (nm)")
fig.suptitle("Fig. Visible emission lines of five elements. Each element has its own fingerprint.",
             fontsize=10)
fig.tight_layout()

Spectral lines and Rydberg’s formula¶

• The existence of discrete spectral lines is impossible to describe with classical mechanics. In 1885, Johann Balmer demonstrated that a subset of the hydrogen atom spectrum (the Balmer series) could be described by the equation

where $n=3,4,5,...$. Written in terms of the wavenumber $\tilde{\nu} = 1/\lambda$ this is $\tilde{\nu} = \frac{4}{B}\left(\frac{1}{2^2}-\frac{1}{n^2}\right)$ with $4/B = 1.097\times10^{7}\ \text{m}^{-1}$. Later, Johannes Rydberg generalized this formula to account for the entire hydrogen atom spectrum yielding the Rydberg formula

• While these equations fit the hydrogen atom spectrum nicely, they do not prescribe any physics to the system. They do not present a model of the hydrogen atom but rather a heuristic equation that fits the data. Nonetheless, scientists were perplexed by the presence of the integers $n_1$ and $n_2$.

Bohr’s Model of the Hydrogen Atom¶

• In 1913, Niels Bohr proposed a model of the hydrogen atom that successfully explained its discrete emission spectrum.

• The atom was pictured as an electron moving in circular orbits around a central proton. Because the proton is far more massive than the electron, it was treated as fixed in space.

• To prevent the electron from spiraling into the nucleus, Bohr introduced a new quantization rule: the electron’s orbital motion must accommodate an integer number of standing wave modes, $n = 1, 2, 3, \ldots$

• This postulate leads directly to an expression for the allowed energy levels of hydrogen, each labeled by a principal quantum number $n$.

Quantizing the States of the Electron in the Hydrogen Atom¶

• Imposing this condition gives the relation

• Here, $\lambda_e$ is the de Broglie wavelength of the electron:

• Substituting this expression for $\lambda_e$ into the quantization condition yields

• We introduce the shorthand $\hbar = \tfrac{h}{2\pi}$ because it appears frequently in quantum mechanics. The left-hand side, $m_e v r$, represents the angular momentum of the electron.

• Thus, Bohr’s model predicts that the electron’s angular momentum is quantized in integer multiples of $\hbar$.

Force Balance¶

After introducing his quantization rule, Bohr turned back to classical mechanics to determine the allowed electron energies. He assumed that, in a stationary orbit, the electrostatic attraction between the proton and electron is exactly balanced by the centrifugal force of the orbiting electron.

Electrostatic force

where $e$ is the elementary charge and the factor $4\pi \varepsilon_0$ ensures SI units.

Centrifugal force

where $m_e$ is the electron mass and $v$ its orbital velocity.

Equating these two forces gives

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from matplotlib.patches import Circle
from IPython.display import HTML

TEAL, CARDINAL, GRAY, PURPLE = "#107895", "#C8102E", "#6c757d", "#6a3d9a"
R2, N_FR, OM = 1.0, 72, 1.0           # radius, frames, angular velocity
SV, SA = 0.62, 0.48                   # arrow scale factors
th_seq = np.linspace(0, 2 * np.pi, N_FR, endpoint=False)

fig_cm, (cx, dx) = plt.subplots(1, 2, figsize=(10.4, 4.5),
                                gridspec_kw={"width_ratios": [1.15, 1.2]})

cx.add_patch(Circle((0, 0), R2, fill=False, color=GRAY, lw=1.2, ls="--"))
cx.plot(0, 0, "o", color=CARDINAL, ms=11)
cx.text(0.10, -0.22, "nucleus", color=CARDINAL, fontsize=8)
(bead,) = cx.plot([], [], "o", color="k", ms=9, zorder=6)

def arrow(color, ls="-", lw=2.2):
    return cx.annotate("", xy=(0, 0), xytext=(0, 0), zorder=5,
                       arrowprops=dict(arrowstyle="-|>", color=color, lw=lw,
                                       ls=ls, shrinkA=0, shrinkB=0))

a_r, a_v, a_a, a_cf = arrow(GRAY, lw=1.3), arrow(TEAL), arrow(PURPLE), arrow(CARDINAL, "--")
a_r.arrow_patch.set_alpha(0.45)
cx.set_xlim(-1.75, 1.75)
cx.set_ylim(-1.6, 1.6)
cx.set_aspect("equal")
cx.axis("off")
cx.legend(handles=[plt.Line2D([], [], color=GRAY, lw=1.3, alpha=0.45, label=r"$\vec{r}$  position"),
                   plt.Line2D([], [], color=TEAL, lw=2.2, label=r"$\vec{v}$  velocity (tangent)"),
                   plt.Line2D([], [], color=PURPLE, lw=2.2, label=r"$\vec{a}$  centripetal (inward)"),
                   plt.Line2D([], [], color=CARDINAL, lw=2.2, ls="--",
                              label="centrifugal (rotating frame)")],
          loc="lower center", bbox_to_anchor=(0.5, -0.16), fontsize=7.5, frameon=False)

t_full = th_seq / OM
dx.plot(t_full, R2 * np.cos(th_seq), color=GRAY, lw=1.6, label=r"$x = R\cos\omega t$")
dx.plot(t_full, -OM * R2 * np.sin(th_seq), color=TEAL, lw=1.6, label=r"$v_x = -\omega R\sin\omega t$")
dx.plot(t_full, -OM**2 * R2 * np.cos(th_seq), color=PURPLE, lw=1.6,
        label=r"$a_x = -\omega^2 R\cos\omega t$")
sweep = dx.axvline(0, color="k", lw=0.9, alpha=0.45)
(m_x,) = dx.plot([], [], "o", color=GRAY, ms=7)
(m_v,) = dx.plot([], [], "o", color=TEAL, ms=7)
(m_a,) = dx.plot([], [], "o", color=PURPLE, ms=7)
dx.axhline(0, color="k", lw=0.6, alpha=0.3)
dx.set_xlim(0, t_full[-1])
dx.set_ylim(-1.5, 1.9)
dx.set_xlabel("time")
dx.set_yticks([])
dx.legend(loc="upper right", fontsize=7.5, frameon=False)
dx.set_title("each component is simple harmonic motion", fontsize=9)
for sp in ("top", "right", "left"):
    dx.spines[sp].set_visible(False)

fig_cm.suptitle("Fig. Circular motion: position, velocity and centripetal acceleration.\n"
                "The magnitudes never change, only the directions.", fontsize=10)
fig_cm.tight_layout()

def frame_cm(i):
    th = th_seq[i]
    p = np.array([R2 * np.cos(th), R2 * np.sin(th)])
    rhat, that = p / R2, np.array([-np.sin(th), np.cos(th)])
    bead.set_data([p[0]], [p[1]])
    a_r.xy = p
    a_r.set_position((0, 0))
    a_v.xy = p + SV * OM * R2 * that
    a_v.set_position(p)
    a_a.xy = p - SA * OM**2 * R2 * rhat
    a_a.set_position(p)
    a_cf.xy = p + SA * OM**2 * R2 * rhat
    a_cf.set_position(p)
    sweep.set_xdata([t_full[i], t_full[i]])
    m_x.set_data([t_full[i]], [p[0]])
    m_v.set_data([t_full[i]], [-OM * R2 * np.sin(th)])
    m_a.set_data([t_full[i]], [-OM**2 * R2 * np.cos(th)])
    return bead, a_r, a_v, a_a, a_cf, sweep, m_x, m_v, m_a

ani_cm = FuncAnimation(fig_cm, frame_cm, frames=N_FR, interval=50, blit=False)
plt.close(fig_cm)
HTML(ani_cm.to_jshtml())

The force-balance equation together with the quantized angular momentum condition restricts the allowed radii $r$ of electron orbits. Solving step by step:

1. From angular momentum quantization:

   $$m_e v r = n\hbar \quad \Rightarrow \quad v = \frac{n\hbar}{m_e r}.$$

   (9)

2. Substituting into the force-balance equation:

   $$\frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{m_e}{r} \left( \frac{n\hbar}{m_e r} \right)^2.$$

   (10)

3. Simplifying:

   $$\frac{e^2}{4\pi\varepsilon_0} = \frac{(n\hbar)^2}{m_e r}.$$

   (11)

4. Solving for $r$:

   $$r = \frac{4\pi \varepsilon_0 (n\hbar)^2}{m_e e^2} = n^2 a_0, \quad n = 1, 2, 3, \ldots$$

   (12)

• where the constant $a_0$ is the Bohr radius, corresponding to the size of the ground-state orbit.

• We see clearly that the radius of an orbit grows with increasing quantum number $n=1,2,3$.

Energy of the Hydrogen Atom¶

The total energy of the electron-proton system is the sum of the electron’s kinetic energy and the Coulomb potential energy:

Using the force-balance relation

we substitute into the energy expression:

Next, inserting the quantized orbital radius

gives the Bohr energy levels:

Spectral lines and the Rydberg constant¶

The energy difference between two levels $n_1$ and $n_2$ is

Relating this to photon energy $E = h\nu$ and the wavenumber $\tilde{\nu} = \nu/c$ gives

where $R_H$ is the Rydberg constant, which we now know expressed in fundamental constants rather than obtained as the result of an experimental fit!

Hydrogen-like atoms¶

• For one-electron atoms such as $He^{+}$ and $Li^{2+}$, the Bohr model still works, but we have to account for the increased nuclear charge $Z$.

• E.g. for the $H$ atom $Z=1$, for $He^{+}$ $Z=2$, etc.

Explore the hydrogen spectrum¶

Every spectral series is the set of transitions that end on one lower level. Slide $n_1$ to move from the Lyman series (ultraviolet) through Balmer (the visible lines of a hydrogen lamp) to Paschen and Brackett (infrared). Raise $Z$ to see how a one-electron ion like $He^+$ pulls all levels down by $Z^2$ and pushes every line to shorter wavelengths.

Where Bohr’s model breaks down¶

Bohr’s model reproduces the hydrogen spectrum to four digits, and that success is exactly why its failures matter. Within a decade it was clear that the model was a lucky halfway house rather than the final theory.

• It fails for every atom with more than one electron. Applied to neutral helium it misses the measured ionization energy of 24.6 eV badly, and it says nothing at all about the periodic table.

• It predicts where lines are but not how bright. Real spectra have strong lines, weak lines, and transitions that never appear at all. Bohr’s rules give no intensities and no selection rules.

• It cannot see fine structure. At high resolution each Balmer line splits into closely spaced components, and a magnetic field splits them further (the Zeeman effect). A single quantum number $n$ has no room for this.

• It assumes an orbit at all. A definite radius together with a definite speed violates the uncertainty principle from the previous lecture. The ground state of hydrogen in fact has zero orbital angular momentum, not $\hbar$.

What survives is the physics, not the picture: energies are quantized, the quantum number is an integer, and light is emitted when the atom drops from one level to another. Chapter 5 replaces the orbit with a wavefunction and gets the levels right for the right reason, Chapter 6 supplies the missing selection rules, and Chapter 7 takes on helium.

Problems¶

Problem 1: Lyman alpha¶

The so-called Lyman series of lines in the emission spectrum of hydrogen corresponds to transitions from various excited states to the n = 1 orbit. Calculate the wavelength of the lowest-energy line in the Lyman series to three significant figures. In what region of the electromagnetic spectrum does it occur?

Problem 2: Photon from n = 4 to n = 1¶

• A. Calculate the energy of a photon that is produced when an electron in a hydrogen atom goes from an orbit with $n=4$ to an orbit with $n=1$.

• B. What happens to the energy of the photon as the initial value of $n$ approaches infinity?

Problem 3: First lines of the Lyman series¶

Use Rydberg’s formula to calculate the first few lines of the Lyman series ($n_1=1$).

Problem 4: Which level did the electron come from?¶

A line in the Lyman series of hydrogen has a wavelength of $1.03 \cdot 10^{-7} m$. Find the original level of the electron.

Problem 5: Ionization energy of He+¶

Using Bohr theory calculate the ionization energy of singly ionized helium $He^{+}$.

Problem 6: Bohr radii¶

• Calculate the radii of the Bohr orbits for the first few levels.

• (Optional) Using python plot $r_n$ vs $n$

Problem 7: The color of H-alpha¶

The brightest visible line of hydrogen, H-alpha, is the $n = 3 \to 2$ transition of the Balmer series. Compute its wavelength and name its color. Do the same for $n = 4 \to 2$ (H-beta). These two lines are what you see in a hydrogen discharge tube, and they give emission nebulae their red glow.

Problem 8: A coincidence between He+ and H¶

Show that the $n = 4 \to 2$ transition of $He^+$ emits a photon of exactly the same energy as the $n = 2 \to 1$ (Lyman alpha) transition of hydrogen. Find the general rule: which $He^+$ transitions coincide with hydrogen lines, and why?

Problem 9: How fast is the electron?¶

Using $m_e v r = n\hbar$ and $r = n^2 a_0$, find the speed of the electron in the ground state of hydrogen and express it as a fraction of the speed of light. This dimensionless ratio is the fine-structure constant $\alpha \approx 1/137$. What does it say about the need for relativity in hydrogen, and what happens to the innermost electron of uranium ($Z = 92$)?

Problem 10: The edge of a series¶

Every spectral series has a longest wavelength (its first line) and a shortest (the series limit, $n_2 \to \infty$). Compute both for the Paschen series ($n_1 = 3$). In which region of the electromagnetic spectrum do they fall, and can the Paschen lines ever overlap with the Balmer lines?